# tetrahedron_nco_rule

tetrahedron_nco_rule, a C++ code which defines the weights and abscisass for a sequence of 7 Newton-Cotes open quadrature rules over the interior of a tetrahedron in 3D.

Newton-Cotes rules have the characteristic that the abscissas are equally spaced. For a tetrahedron, this refers to spacing in the unit reference tetrahedron, or in the barycentric coordinate system. These rules may be mapped to an arbitrary tetrahedron, and will still be valid.

The rules are said to be "open" when they do not include points on the boundary of the tetrahedron.

The use of equally spaced abscissas may be important for your application. That may how your data was collected, for instance. On the other hand, the use of equally spaced abscissas carries a few costs. In particular, for a given degree of polynomial accuracy, there will be rules that achieve this accuracy, but use fewer abscissas than Newton-Cotes. Moreover, the Newton-Cotes approach almost always results in negative weights for some abscissas. This is generally an undesirable feature, particularly when higher order quadrature rules are being used.

### Languages:

tetrahedron_nco_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version

### Related Data and Programs:

LINE_NCO_RULE, a C++ code which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

TETRAHEDRON_ARBQ_RULE, a C++ code which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

TETRAHEDRON_EXACTNESS, a C++ code which investigates the monomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

TETRAHEDRON_FELIPPA_RULE, a C++ code which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_INTEGRALS, a C++ code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_KEAST_RULE, a C++ code which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_MONTE_CARLO, a C++ code which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_NCC_RULE, a C++ code which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_NCO_RULE, a C++ code which defines Newton-Cotes open quadrature rules over the interior of a triangle in 2D.

### Reference:

1. Peter Silvester,